Problem: Logan genetically engineered a new type of fir tree and a new type of pine tree. The combined height of one fir tree and one pine tree is $21$ meters. The height of $4$ fir trees stacked on top of each other is $24$ meters taller than one pine tree. How tall are the types of trees that Logan genetically engineered? Each fir tree is
Let $x$ represent the height of a fir tree and let $y$ represent the height of a pine tree. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that the combined height of one fir tree and one pine tree is $\textit {21}$ meters. How can we model this sentence algebraically? Since the heights of a fir tree and pine tree add up to $21$ meters, we get the following equation: $x+ y = 21$ We are also given that the combined height of $\textit{4}$ fir trees is $\textit{24}$ meters more than the height of a single pine tree. This can be expressed as: $y +24 = 4x$ Let's rewrite this equation so that it's solved for $y$ : $y = 4x-24$ Now that we have a system of two equations, we can go ahead and solve it! Let's substitute $ y={4x-24}$ into the first equation: $\begin{aligned}x+ y &= 21\\\\ x+({4x-24})&=21\\\\ 5x &=45\\\\ x&=9\end{aligned}$ Now we can substitute $x = 9$ into $x+y=21$ and find that $y=12$. Recall that $x$ denotes the height of each fir tree and $y$ denotes the height of each pine tree. Therefore, each fir tree is $\textit{9}$ meters tall and each pine tree is $\textit{12}$ meters tall.